Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 7/13/2023
Public

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?hesv

Computes the solution to the system of linear equations with a Hermitian matrix A and multiple right-hand sides.

Syntax

call chesv( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info )

call zhesv( uplo, n, nrhs, a, lda, ipiv, b, ldb, work, lwork, info )

call hesv( a, b [,uplo] [,ipiv] [,info] )

Include Files

  • mkl.fi, lapack.f90

Description

The routine solves for X the complex system of linear equations A*X = B, where A is an n-by-n symmetric matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.

The diagonal pivoting method is used to factor A as A = U*D*UH or A = L*D*LH, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

The factored form of A is then used to solve the system of equations A*X = B.

Input Parameters

uplo

CHARACTER*1. Must be 'U' or 'L'.

Indicates whether the upper or lower triangular part of A is stored and how A is factored:

If uplo = 'U', the array a stores the upper triangular part of the matrix A, and A is factored as U*D*UH.

If uplo = 'L', the array a stores the lower triangular part of the matrix A, and A is factored as L*D*LH.

n

INTEGER. The order of matrix A; n 0.

nrhs

INTEGER. The number of right-hand sides, the number of columns in B; nrhs 0.

a, b, work

COMPLEX for chesv

DOUBLE COMPLEX for zhesv.

Arrays: a(size lda by *), bb(size ldb by *), work(*). The array a contains the upper or the lower triangular part of the Hermitian matrix A (see uplo). The second dimension of a must be at least max(1, n).

The array b contains the matrix B whose columns are the right-hand sides for the systems of equations. The second dimension of b must be at least max(1,nrhs).

work is a workspace array, dimension at least max(1,lwork).

lda

INTEGER. The leading dimension of a; lda max(1, n).

ldb

INTEGER. The leading dimension of b; ldb max(1, n).

lwork

INTEGER. The size of the work array (lwork 1).

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla. See Application Notes below for details and for the suggested value of lwork.

Output Parameters

a

If info = 0, a is overwritten by the block-diagonal matrix D and the multipliers used to obtain the factor U (or L) from the factorization of A as computed by ?hetrf.

b

If info = 0, b is overwritten by the solution matrix X.

ipiv

INTEGER.

Array, size at least max(1, n). Contains details of the interchanges and the block structure of D, as determined by ?hetrf.

If ipiv(i) = k > 0, then dii is a 1-by-1 diagonal block, and the i-th row and column of A was interchanged with the k-th row and column.

If uplo = 'U'and ipiv(i) =ipiv(i-1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i-1, and (i-1)-th row and column of A was interchanged with the m-th row and column.

If uplo = 'L'and ipiv(i) =ipiv(i+1) = -m < 0, then D has a 2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column of A was interchanged with the m-th row and column.

work(1)

If info = 0, on exit work(1) contains the minimum value of lwork required for optimum performance. Use this lwork for subsequent runs.

info

INTEGER. If info = 0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could not be computed.

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or reconstructible arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine hesv interface are as follows:

a

Holds the matrix A of size (n,n).

b

Holds the matrix B of size (n,nrhs).

ipiv

Holds the vector of length n.

uplo

Must be 'U' or 'L'. The default value is 'U'.

Application Notes

For better performance, try using lwork = n*blocksize, where blocksize is a machine-dependent value (typically, 16 to 64) required for optimum performance of the blocked algorithm.

If you are in doubt how much workspace to supply, use a generous value of lwork for the first run or set lwork = -1.

If you choose the first option and set any of admissible lwork sizes, which is no less than the minimal value described, the routine completes the task, though probably not so fast as with a recommended workspace, and provides the recommended workspace in the first element of the corresponding array work on exit. Use this value (work(1)) for subsequent runs.

If you set lwork = -1, the routine returns immediately and provides the recommended workspace in the first element of the corresponding array (work). This operation is called a workspace query.

Note that if you set lwork to less than the minimal required value and not -1, the routine returns immediately with an error exit and does not provide any information on the recommended workspace.